Joint Probability

Author: nitin.p070

Probability in math is defined as the chance of happening something in future. The two random variables A and B are defined on the same probability space, the joint probability distribution for A and B defines the probability of events defined in terms of both A and B. In the case of having only two random variables, this is called a bi-variate distribution, but the thought simplify to any number of random variables, giving a multivariate distribution.

Joint probability:

A numerical measure where the chance of two events happening together and at the same time are calculated. For example if the probability of event B happening at the same time event A happens, then the Joint probability has been given as follows.
Joint probability notation takes the form:

P (A `nn` B) or P (A and B)

Indicates the joint probability of A and B.

Example: The joint probability can be calculated by rolling a 2 and a 5 with two dissimilar dice.

with and without replacemant-Joint probability:

Joint probability is used in multistage testing .Joint probability can be done with replacement or without replacement.

With replacement: It indicates that the thing that are chosen on one stage are returned to the sample space before the next choice is made .For example, tossing a head on the first toss does not affect the outcome of flipping the coin a second time.

The probability that independent events A and B occur at the same time can be found by using the multiplication rule, or the product of the entity probabilities.

Example 1:

If you pick two cards from the deck without replacement, find the probability that they will both be aces.

Solution:

Total number of aces in the deck of cards = 4.

Cards picked up = 2 aces.

total number of aces* (total number of aces-1)
Hence the probability = --------------------------------------------------
total number of cards*( total number of cards-1)

P (AA) = `(4/52)*(3/51)` = `1/221` .

Without replacement:

Uses the same idea, if the first choice is not replaced only we consider the change in the sample space. Still we use the multiplication rule, but for each of the stages the numerator and/or denominator decreases

Example 2:

Find the probability of tossing a fair coin twice in a row, getting heads both times.

Solution:

While tossing a fair coin once we get head or tail.

Given that while tossing a coin head occurs,So

We know that the probability =(Number of favourable outcomes/Total number of outcomes)

Therefore probability of getting head while tossing the coin once P(H)= `1/2.`

Similarly tossing a coin next time we have probability of getting head P(H) =`1/2.`

As the question is to find the probability of tossing a fair coin twice in a row, getting heads both times it indicates that we have to find the joint probability without replacement.

As the probability of tossing a head is ` 1/2 ` each time P (H,H) =` (1/2) *(1/2) = 1/4.`

Article Source: http://www.articlesbase.com/science-articles/joint-probability-6618060.html

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